Maths - Plücker Coordinates

Ways to specify a Line in 3D space

If a line goes through the origin we can specify it using a single vector but how do we specify a line that does not go though the origin?

Here are some possibilities:

If the line is a finite length we can specify the endpoints of the line.

If the line is of infinite length we can specify any two points on the line.

Or we can specify one point on the line and the direction of the line.

These methods all require 2 vectors (6 scalar values) to specify the line.

The disadvantage of these methods is that they involve one or two arbitrary points. Therefore two identical lines may be specified differently and so we cant easily check if the lines are identical or how close they are to each other.

A better way to do this would be to use the direction of the line and the vector direction from the origin to the nearest point on the line to the origin.

A related way is to start with an arbitrary point on the line and take its cross product with the direction vector. This cross product is independent of the point chosen and uniquely defines the line which is what we are looking for. The direction vector along with this cross product are known as plücker coordinates and are denoted by:

(v ; P × v)

The semicolon ';' distinguishes it from a 6 dimensional vector.

where:

v : unit vector (normalised to unit length)

P × v : known as the 'moment' its direction is mutually perpendicular to both the vector and to the shortest distance between the origin and the line.

Intersecting Lines

If we have two lines, shown here in blue, which are not necessarily parallel or related to each other.

There are many possible lines, example shown in red, which intersect the two lines.

If we increase the number of lines to three there may still be many possible solutions. If we increase the number of lines to four there may be 2 solutions in the most general case. These solutions are given by a quadratic equation.

Example

Calculate the plücker coordinates of the following lines:

A : A line form (5,1,0) to (1,5,0) with a rotation of 30° around the line (-0.7071,0.7071,0)

B : A line through the point (4,1,0) in the direction (-0.7071,0.7071,0) with a rotation of 30° around the line (-0.7071,0.7071,0)

C : A line form (-1,2,5) to (6,4,-2) with a rotation of 60° around the line (0,0.7071,0.7071)

Applications

One important application of plucker coordinates concerns the transform of a solid body in mechanics which may include both rotation and translation. This can be done using screw theory which often uses plucker coordinates to define the line of the screw.

Screw Theory

We can represent any movement of a solid body by a single operation which combines both the rotation and the translation. This page explains this.

Rays

A force on a solid body has a magnitude and direction, but it also the position where the force acts (discussed on this page).

A ray of light has a direction and a magnitude (brightness) and also a starting point where the light emanates from (discussed on this page).

Can we represent such quantities as a single mathematical element which can represent both vectors in a single equation? The vectors which make up these seem to be true vectors (not bivectors). Can we convert the two vectors into one vector and a bivector? For example, all the forces on a body can be resolved into a single force together with a torque.

Rays

Some things require 2 vectors to define them, for example,

A force on a solid body has a magnitude and direction, but it also the position where the force acts (discussed on this page).

A ray of light has a direction and a magnitude (brightness) and also a starting point where the light emanates from (discussed on this page).

Can we represent such quantities as a single mathematical element which can represent both vectors in a single equation? The vectors which make up these seem to be true vectors (not bivectors). Can we convert the two vectors into one vector and a bivector? For example, all the forces on a body can be resolved into a single force together with a torque.

Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.

New Foundations for Classical Mechanics (Fundamental Theories of Physics). This
is very good on the geometric interpretation of this algebra. It has lots of insights
into the mechanics of solid bodies. I still cant work out if the position, velocity,
etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors
are required to represent translation and rotation.